FFT delay: Catch 22

Summary

It is impossible to delay [...-1, 1, -1, 1, ...] by a fraction of a sample, no matter how hard you try

When using an even sized Fourier transform (FFT), the frequency at the Nyquist limit has no phase information.
Therefore, it cannot be phase shifted, except trivially in steps of one sample.
In other words, the task of delaying the signal is impossible.

The experiment below shows what happens, if one tries nonetheless:

Figure 1 shows a test signal (black impulses), that was delayed using FFT methods.
The red curve is the continuous-time signal reconstructed from the delayed samples.
The blue curve shows the equivalent continuous-time signal, time-shifted by the same amount (ideal result).
For an odd number of points, red and blue curve match for all delays.

Figure 1: Delaying an odd-sized signal

Figure 2 shows the same experiment, this time with an even number of points.
This time, red and blue curve match only for the undelayed data, and for the final set of points with a delay of one full sample.

Delaying an even-sized signal

© Markus Nentwig 2007-2008
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